A note on Mobius transformations and Bogolubov coefficients A. Fabbri ,

A note on Mobius transformations
and Bogolubov coefficients1
A. Fabbria,b ,
2
J. Navarro-Salasb
3
and G. Olmob,c
4
a) Dipartimento di Fisica dell’Università di Bologna and INFN sezione di Bologna, Via Irnerio
46, 40126 Bologna, Italy
b) Departamento de Fı́sica Teórica and IFIC, Centro Mixto Universidad de Valencia-CSIC.
Facultad de Fı́sica, Universidad de Valencia, Burjassot-46100, Valencia, Spain
c) Physics Department, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA
Abstract
We analyze the creation of scalar massless particles in two dimensions under the action
of conformal transformations. We focus our attention to Mobius transformation and clarify
an apparent tension between the results obtained with the Bogolubov coefficients and those
obtained within the conformal field theory approach.
1
To be published in the proceedings of ” Symmetries in Gravity and Field Theory”, Work-
shop in honour of Prof. J. A. de Azcarraga. June 9-11, 2003. Salamanca (Spain).
2
3
4
e-mail: [email protected]
[email protected]
[email protected]
One of the basic ingredients of quantum field theory in curved spacetime
[1] are the Bogolubov transformations. These reflect the absence, in general, of
a privileged vacuum state, in parallel to the absence of global inertial frames.
This framework is general and can be applied to a large number of physical
situations, including flat spacetime (like the Unruh-Fulling effect [1]). On the
other hand, of particular physical interest are those field theories possessing
the spacetime conformal symmetry SO(d, 2), where d is the dimension of the
Lorentzian spacetime. This symmetry is especially powerful in two dimensions,
where the group SO(2, 2) can be enlarged to an infinite-dimensional group [2].
However, this SO(2, 2) subgroup, which includes dilatations, Poincaré and special conformal transformations, still plays an important role because it leaves
the vacuum invariant [2]. From the point of view of Bogolubov transformations
this should imply that the β coefficients associated to them vanish. This is obvious for Poincaré and dilatations: they do not produce any mixing of positive
and negative frequencies. For special conformal transformations the β coefficients (calculated for a massless scalar field) are vanishing only if one considers
the two branches of the transformation. If we consider only one branch we
get a nonvanishing result [3, 1], which has been interpreted with the fact that
the absence of energy (always true irrespective of the presence of one or two
branches) does not imply absence of quanta too. The purpose of this note is to
clarify this apparent tension between both approaches.
Let us first briefly review the definition of the Bogolubov coefficients for the
two-dimensional massless scalar field f satisfying the wave equation
∇2 f = 0 .
(1)
In conformal gauge ds2 = −e2ρ dx+ dx− we can decompose the field into positive
and negative frequencies using the standard mode solutions:
Z
f=
0
∞
dw → −iwx− ← −iwx+ →† iwx− ←† iwx+
√
( a we
+ a we
+ a we
+ a we
).
4πw
(2)
These modes form an orthonormal basis under the scalar product
(f1 , f2 ) = −i
Z
Σ
dΣµ (f1 ∂µ f2∗ − ∂µ f1 f2∗ ) ,
1
(3)
where Σ is an appropiate Cauchy hypersurface. One can construct the Fock
space from the commutation relations
→
→†
←
←†
[ a w , a w0 ] = δ(w − w0 ) ,
(4)
[ a w , a w0 ] = δ(w − w0 ) .
(5)
The vacuum state |0x i is defined by
→
a w |0x i
←
a w |0x i
= 0,
=0,
(6)
and the excited states can be obtained by the application of creation operators
→† ←†
a w, a w
out of the vacuum. We can perform an arbitrary conformal transfor-
mation
x± → y ± = y ± (x± ) ,
(7)
and consider the expansion
Z
∞
f=
0
dw → −iwy− ← −iwy+ →† iwy− ←† iwy+
√
( b we
+ b we
+ b we
+ b we
).
4πw
As both sets of modes are complete, the new modes
(8)
−
+
√ 1 e−iwy , √ 1 e−iwy
4πw
4πw
can be expanded in terms of the old ones:
±
e−iwy
√
=
4πw
∞
Z
0
dw0 ± −iw0 x±
±
iw0 x±
√
αww0 e
+ βww
e
,
0
4πw0
(9)
±
±
where αww
0 and βww 0 are called the Bogolubov coefficients. These coefficients
can be evaluated by the following scalar products
where φ±
w =
±
−iwx
e√
4πw
±
αww
0
±
βww
0
, φ̄±
w =
1
=
2π
r
1
=−
2π
±
αww
0
±
= (φ̄±
w , φw 0 )
(10)
±
βww
0
±∗
= −(φ̄±
w , φw 0 )
(11)
±
−iwy
e√
4πw
w
w0
r
Z
w
w0
Z
. The results are:
dx±
±
dx
dy ±
dx±
!
dy ±
dx±
2
e−iwy
± (x± )+iw 0 x±
!
e−iwy
± (x± )−iw 0 x±
,
(12)
.
(13)
The relation between creation and annhilation operators in the two basis is
→
bw
∞
Z
dw
=
0
0
−∗ →
αww
0 a w0
−
−∗ →†
βww
0a
,
(14)
→†
along with the corresponding one for b w . Similar equations hold for the left
movers. Therefore the expectation value of the (right mover sector) particle
→
→† →
number operator N w ≡ b w b w is given by the expression
→
h0x |N w |0x i =
Z
0
∞
− 2
dw0 |βww
0| .
(15)
→
Let us now discuss how to obtain an expression for h0x |N w |0x i within
the framework of conformal field theory. The two-point correlation function
h0x |f (x)f (x0 )|0x i is ill-defined due to the infrared divergence of the scalar field
in two dimensions. This can be cured by introducing a frequency cut-off λ. In
doing so one gets
h0x |f (x)f (x0 )|0x i = −
1 2γ + ln λ2 (x − x0 )2 ,
4π
(16)
where γ is the Euler constant. The ambiguity inherent to the cut-off disappears when one considers, instead of the two-point function for the field f , the
correlations for the derivatives ∂± f . We have then
h0x |∂± f (x± )∂± f (x0± )|0x i = −
1
1
.
±
4π (x − x0± )2
(17)
Under conformal transformations x± → y ± = y ± (x± ), the above correlation
functions transform according to the rule for primary fields:
1
h0x |∂± f (y )∂± f (y )|0x i = −
4π
±
0±
dx± (y ± )
dy ±
!
dx± (y 0± )
dy ±
!
1
.
(x± (y ± ) − x0± (y ± ))2
(18)
These relations are fundamental to construct the normal ordered stress tensor
: T±± : as well as the particle number operator. In the coordinates {x± } and the
choice of modes φ±
w , the normal ordered stress tensor operator can be defined
via point-splitting
: T±± (x± ) :=
lim ∂± f (x± )∂± f (x0± ) +
x± →x0±
3
1
1
.
±
4π (x − x0± )2
(19)
Similar relations hold in the coordinates {y ± } and with the choice of the modes
±
±
φ̄±
w . It is easy to relate : T±± (y ) : with : T±± (x ) : and the result is
dx±
dy ±
±
: T±± (y ) :=
!2
: T±± (x± ) : −
1
{x± , y ± } ,
24π
where
d3 x± dx± 3
{x , y } = ±3 / ± −
dy
dy
2
±
d2 x± dx±
/
dy ±2 dy ±
±
(20)
!2
(21)
is the Schwarzian derivative. The two-point correlation function ∂± f (y ± )∂± f (y 0± )
also serves to construct the particle number operator. We start from the explicit form of the normal ordered operator : ∂± f (x± )∂± f (x0± ) : in terms of the
creation and annihilation operators (for simplicity we shall consider only the
right mover sector)
−
0−
h0x | : ∂− f (y )∂− f (y ) : |0x i =
−iw0 y − +iwy 0−
e
)−
Z
∞
Z
√
∞
dw
dw
0
0
→ →
−
0 0−
h0x | b w b w0 |0x ie−iwy −iw y
0
→† →
ww0
−
0 0−
{h0x | b w b w0 |0x i(eiwy −iw y +
4π
→† →†
− h0x | b w b w0 |0x ieiwy
− +iw 0 y 0−
}.
(22)
Now, instead of taking the limit x± → x0± , as in the construction of the stress
tensor, we shall take the following Fourier transform
1
(2π)2
Z
+∞
−∞
√
−
0−
−
−iw̃y − +iw̃0 y 0− )
0−
dy dy h0x | : ∂− f (y )∂− f (y ) : |0x ie
=
→† →
w̃w̃0
h0x | b w̃ b w̃0 |0x i .
4π
(23)
Note that to obtain this last equation it is crucial to integrate over all range
in the coordinates y − , y 0− . We then immediately get an expression for the
→
→† →
expectation value of the particle number operator N w = b w b w of frequency w
:
→
h0x |N w |0x i =
1
πw
Z
+∞
−∞
dy − dy 0− h0x | : ∂− f (y − )∂− f (y 0− ) : |0x ie−iw(y
− −y 0− )
.
(24)
From the above considerations we obtain
→
h0x |N w |0x i = −
"
1
4π 2 w
Z
+∞
−∞
!
−
dx (y − )
dy −
dy − dy 0− e−iw(y
dx− (y 0− )
dy −
4
!
− −y 0− )
1
1
− −
−
0−
2
(x − x )
(y − y 0− )2
(25)
#
.
We mention that the integral
R
R∞
0
→
dwwh0x |N w |0x i gives the integrated flux
dy − h0x | : T−− (y − ) : |0x i.
As an illustrative example we shall show how the CFT approach reproduces
the thermal properties associated to the conformal transformation
±
x± = ±κ−1 e±κy .
(26)
We can think of this transformation as relating the Minkowskian x± and Rindler
y ± null coordinates, where κ is the acceleration parameter (the same relation
holds for the Schwarzschild black hole between the Kruskal and EddingtonFinkelstein null coordinates with κ = 1/4M ). Since we are using plane waves
→† →
instead of wave-packets we first work out an expression for h0x | b w b w0 |0x i
→† →
h0x | b w b w0 |0x i
1
√
= −
−
Z
+∞
dy − dy 0− [
dx− − dx− 0−
1
(y ) − (y ) −
−
dy
dy
(x − x0− )2
4π 2 ww0 −∞
1
−
0 0−
]e−iwy +iw y .
−
0−
2
(y − y )
(27)
Substitution of the relations (26) leads to
→† →
h0x | b w b w0 |0x i
1
=−
δ(w − w0 )
2πw
Z
+∞
−∞
"
#
κ2 e−κz
1
dz
− 2 e−iwz ,
−κz
2
(1 − e )
z
(28)
where z = y − − y 0− . Evaluation of the integral gives
→† →
h0x | b w b w0 |0x i = δ(w − w0 )
1
e
2πw
κ
−1
,
(29)
leading to the number of particles emitted per unit time of
1
→
h0x |N w |0x i =
e
2πw
κ
−1
,
(30)
which corresponds to the Planckian spectrum of radiation at the temperature
T =
κ
2π .
Similar results hold for the left mover sector. Evaluation of the
expectation value of the stress tensor using (20), taking into account that h0x | :
T±± (x± ) : |0x i = 0, gives
h0x | : T±± (y ± ) : |0x i =
5
πT 2
κ2
=
.
48π
12
(31)
This is nothing else but the stress tensor corresponding to a two dimensional
thermal bath of radiation at the temperature T .
We shall now analyse the case associated to the Mobius transformations
x± → y ± =
a± x± + b±
c± x± + d±
(32)
where a± d± − b± c± = 1. These form the so called global conformal group
((SL(2, R) ⊗ SL(2, R))/Z2 ≈ SO(2, 2)) and have the property of giving a
vanishing Schwarzian derivative. Therefore, under the action of the Mobius
transformations the flux of radiation in the vacuum |0x i for the observer {y ± }
vanishes
h0x | : T±± (y ± ) : |0x i = 0 .
(33)
Moreover, since the two-point function (17) is invariant under (32) it is clear
from (25) that the expectation value of the particle number operator also vanishes
→
←
h0x |N w |0x i = 0 = h0x |N w |0x i .
(34)
This is what we expect in the context of Conformal Field Theory, since the
vacuum is invariant under Mobius transformations. However, the conclusion
is different in the approach of the Bogolubov coefficients. For those Mobius
transformations which are not dilatations nor Poincaré such as
x− = −
1
a2 y −
,
(35)
where a is an arbitrary constant, the Bogolubov coefficients are
αww0
βww0
+∞
0
w
−
0
2 −
− −iwy − −iw0 /a2 y −
=
dy
e
+
dy − e−iwy −iw /a y
,
w0
0
−∞
r
Z +∞
Z 0
1
w
−
0
2 −
− −iwy − +iw0 /a2 y −
= −
dy
e
+
dy − e−iwy +iw /a y (36).
0
2π w
0
−∞
1
2π
r
Z
Z
If one restricts only to one branch (for instance 0 < y − < +∞) the results,
given in [3, 1, 4], are
αww0
=
βww0
=
q
1
K1 (2i ww0 /a2 ) ,
aπ
q
i
K1 (2 ww0 /a2 ) ,
aπ
6
(37)
where K1 is a modified Bessel function. Therefore confining to just one branch
would seem to give paradoxical results, namely a vanishing flux but a production
of quanta due to the nonvanishing β coefficient
→
h0x | N w |0x i =
6 0.
(38)
We mention that the transformation (35) originally appeared in the moving
mirror model of Davies and Fulling [3] and more recently in the analysis of
extremal black holes [5, 6, 7] and in the late-time behaviour of evaporating
near-extremal Reissner-Nordstrom black holes [8]. On the other hand, (38),
based on (37), does not only give a nonvanishing particle number, but also
exhibits a logarithmic infrared divergence. It has been suggested in [5] that such
divergence can be cured, as usual, by using wave packets instead of plane waves.
However, it has been recently pointed out in [6] that the integrals (12), (13)
defining the Bogolubov coefficients are not well defined for the transformation
(35). Indeed, the results (37) are obtained by means of an unjustified Wick
rotation. This problem cannot be cured by using wave packets [6].
The results (38), (37) and (34) are in apparent contradiction. The result
(34) is well established, since the Mobius invariance of the vacuum is almost
an axiom of CFT. It cannot be eluded if one wants to maintain the conformal
invariance at the quantum level. However, the full Mobius transformation (35),
on which the CFT results are based, cannot be restricted only to one branch.
Therefore in order to make the comparison one has to consider also the second
integral, from −∞ to 0, in the formulas (36) leading to
αww0
= Re
βww0
= 0.
q
2
K1 (2i ww0 /a2 ) ,
aπ
(39)
This result restores compatibility with the CFT, as zero β coefficient implies
that no quanta are produced, in agreement with the fact that the energy flux is
zero. This result of the cancellation of the β coefficient should be general and
independent of the prescription one could use to properly define the integral
(36).
7
A.F. and J.N-S thank V. Frolov, S. Gao and A. Zelnikov and G.O. thanks L.
Parker for very useful discussions. This research has been partially supported by
the research grants BFM2002-04031-C02-01 and BFM2002-03681 from the Ministerio de Ciencia y Tecnologia (Spain), EU FEDER funds, the INFN-CYCIT
Collaborative Program and the Generalitat Valenciana. G.J.O. acknowledges
the Department of Physics of the University of Wisconsin at Milwaukee for
hospitality and the Generalitat Valenciana for financial support.
References
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page 1
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8